A pebble is thrown downwards at [tex]$6.70 m / s$[/tex] from a 38.8 m tall bridge and travels in free fall. What is the final velocity of the pebble just before it hits the ground? [tex]$v_{f}=[?] m / s$[/tex]
Do not account for air resistance. Remember, velocity downward is a negative vector (-).
Answer (2)
We are given the initial velocity v i = − 6.70 m / s , height h = 38.8 m , and acceleration due to gravity g = − 9.8 m / s 2 .
Apply the kinematic equation v f 2 = v i 2 + 2 g h .
Substitute the given values: v f 2 = ( − 6.70 ) 2 + 2 ( − 9.8 ) ( − 38.8 ) = 805.37 .
Solve for v f and take the negative root since the pebble is moving downwards: v f = − 805.37 ≈ − 28.379 m / s . The final answer is − 28.379 m / s .
Explanation
Problem Setup and Given Information We are given that a pebble is thrown downwards from a bridge with an initial velocity of v i = − 6.70 m / s . The height of the bridge is h = 38.8 m . We need to find the final velocity v f of the pebble just before it hits the ground, considering free fall and neglecting air resistance. The acceleration due to gravity is g = − 9.8 m / s 2 .
Applying the Kinematic Equation To find the final velocity, we can use the following kinematic equation: v f 2 = v i 2 + 2 g h
Substituting Values Substitute the given values into the equation: v f 2 = ( − 6.70 m / s ) 2 + 2 ( − 9.8 m / s 2 ) ( − 38.8 m ) v f 2 = 44.89 m 2 / s 2 + 760.48 m 2 / s 2 v f 2 = 805.37 m 2 / s 2
Calculating the Final Velocity Now, solve for v f . Since the pebble is traveling downwards, we take the negative square root: v f = 805.37 = − 28.379 m / s Therefore, the final velocity of the pebble just before it hits the ground is approximately -28.379 m/s.
Final Answer The final velocity of the pebble just before it hits the ground is approximately − 28.379 m / s .
Examples
Understanding the final velocity of a falling object is crucial in various real-world scenarios. For instance, engineers designing safety equipment for construction sites need to calculate the impact velocity of falling debris to ensure that safety nets and helmets can withstand the force. Similarly, in sports, knowing the velocity of a ball or projectile at impact helps in designing protective gear and optimizing performance. This problem demonstrates a fundamental physics principle with practical applications in engineering, safety, and sports.
The final velocity of the pebble is calculated using the kinematic equation v f 2 = v i 2 + 2 g h . Substituting the values leads to a final velocity of approximately − 28.379 m / s just before impact. This negative sign indicates the direction of the velocity is downward.
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